What about $x^2 -1?$

Number Theory Level 1

$3^2- 1$ is not a prime number.

$4^2 -1$ is not a prime number.

$5^2-1$ is not a prime number.

$6^2 -1$ is not a prime number.

$7^2 -1$ is not a prime number.

$8^2 -1$ is not a prime number.

Is it true that $x^2 -1$ is never a prime number for all positive integers $x \gt 2$ ?

No Yes

2 solutions

Tarsem Singh Khalsa
Dec 18, 2017

$x^2-1$ can be written as $x^2-1^2$ and can always be factored into

$x^2-1^2 = (x+1)(x-1)$

i.e. there always exist atleast 2 factors ( other than 1 and the no. itself ; which is btw direct voilation of definition of prime no.) for any value $x$ except 2.

If we put $x=2$ then

$x^2-1 = 2^2-1 = (2+1)(2-1) = 3*1$

where 1 and 3 are the only two factors of 3. Hence, $x=2$ gives a prime number.

So, we can say for any value of $x$ , (except 2) $x^2-1$ is not a prime number.

Marta Reece
Dec 18, 2017

You can always factor a difference of squares.

$(2n)^2-1=(2n+1)(2n-1)$

It would actually work for any $x$ , odd or even

$x^2-1=(x+1)(x-1)$

What about x 2 − 1 ? x^2 -1? x 2 − 1 ?

2 solutions

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What about $x^2 -1?$